We call any differential equation of the form $$M(x, y)\,\mathrm{d}x + N(x, y)\,\mathrm{d}y = 0$$ an 'exact differential equation' when $$M_y= N_x.$$
It appears that the following differential equation is not exact, $$(2x + y)\,\mathrm{d}x - (x + 6y)\,\mathrm{d}y = 0$$ because $ M_y = 1 \ne -1 = N_x$. But my friend is insisting that this an 'exact differential equation'. Comment please.
It is not exact; if your friend insists it is, tell them to show you the potential function.
The equation, on the other hand, is homogenous. You have $$ y'=\frac {2x+y}{x+6y}=\frac {2+y/x}{1+6y/x} . $$